Each `matrix' \(\hat{X}_{I=1,\cdots ,9}\)  consists of \(N^2\)  operators, and \(\hat{X}\)  and conjugate momentum \(\hat{P}\)  satisfy the canonical commutation relation.
The states mapped to each other via SU(\(N\) )-transformations are identified.This is equivalent to considering the gauge-invariant Hilbert space. See the following discussion and Ref. [1]}.
Refs. [2]}, [3]}, [4]} assumed the existence of c-number matrices \(X_{I=1,\cdots ,9}\)  that satisfy \({\rm Tr}X_I^2=\langle {\rm Tr}\hat{X}_I^2\rangle \sim N^2\) .
Implicitly, they assumed a coordinate-eigenstate \(|X\rangle \)  that satisfies \(\hat{X}_{I,ij}|X\rangle =X_{I,ij}|X\rangle \)  for all \(I,i,j\) .
This assumption is not legitimate, because low-energy states are not coordinate-eigenstates.
Rather, we should consider wave packets in \(\mathbb {R}^{9N^2}\) .
The `matrices' that carry the information of D-branes and strings are the center of a wave packet.
The location of the wave packet can move in \(\mathbb {R}^{9N^2}\)  via gauge transformation, but the size and shape do not change. The distribution of D-branes determined by the center of the wave packet is well localized. In particular, there is a ground state in which all D-branes are at the origin of \(\mathbb {R}^{9}\) , that is gauge-invariant and localized around the origin of \(\mathbb {R}^{9N^2}\) .
Various well-localized wave packets around different points in \(\mathbb {R}^{9N^2}\)  exist, and the overlap between two wave packets remains small if the centers of wave packets are separated more than an \(O(N^0)\)  value.
<FIGURE>